224
5 Plasticity
Then the stationarity conditions corresponding to Eqs. 5.93a and 5.93b are the constitutive relations
˙
p (σ p ) ∈ d σ p π
∗
(σ p ),
(5.94a)
σ p ( ˙
p ) ∈ d ˙
p π ( ˙
p ).
(5.94b)
Obviously the relations in Eqs. 5.94a and 5.94b determine entirely the dissipative
behavior of the generic Prandtl model, thus the formulation would be completed at
this stage.
To be more explicit, however, alternatively to Eq. 5.94b the closed and convex
admissible domain A in the σ p -space is introduced. It is characterized by the convex
yield condition
φ = φ(σ p ) := ϕ(σ p ) − σ y ≤ 0.
(5.95)
Here φ = φ(σ p ) is the yield function and ϕ(σ p ) denotes the equivalent (plastic) stress
that is compared to the yield limit σ y , a material property. Then the evolution law for
the plastic strain (i.e. the associated flow rule) follows alternatively to Eq. 5.94a from
the postulate of maximum dissipation (due to plasticity) with a Lagrange functional
incorporating the admissibility constraint φ ≤ 0 by the Lagrange multiplier λ ≥ 0
(σ p , λ; ˙
p ) := −d(σ p ; ˙
p ) + λ φ(σ p ).
(5.96)
Consequently, the stationarity condition of this constrained optimization problem
reads
˙
p = λ ∂ σ p φ,
(5.97)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker form
λ ≥ 0, φ ≤ 0, λ φ = 0.
(5.98)
It shall be noted that collectively Eqs. 5.95, 5.97 and 5.98 are entirely equivalent
statements to Eqs. 5.94a and 5.94b.
As a further interesting aspect the dissipation d = σ p ˙
p shall next be examined
more closely. From Eqs. 5.93a and 5.93b the dissipation d is alternatively expressed
in terms of the dissipation potential π and the dual dissipation potential π
∗ as
d = π(˙ p ) + π
∗
(σ p ) ≥ 0.
(5.99)
However, based on the above introduction of the yield condition φ ≤ 0 the dual
dissipation potential is identified as the indicator function I A of the admissible
domain A
5 Plasticity
Then the stationarity conditions corresponding to Eqs. 5.93a and 5.93b are the constitutive relations
˙
p (σ p ) ∈ d σ p π
∗
(σ p ),
(5.94a)
σ p ( ˙
p ) ∈ d ˙
p π ( ˙
p ).
(5.94b)
Obviously the relations in Eqs. 5.94a and 5.94b determine entirely the dissipative
behavior of the generic Prandtl model, thus the formulation would be completed at
this stage.
To be more explicit, however, alternatively to Eq. 5.94b the closed and convex
admissible domain A in the σ p -space is introduced. It is characterized by the convex
yield condition
φ = φ(σ p ) := ϕ(σ p ) − σ y ≤ 0.
(5.95)
Here φ = φ(σ p ) is the yield function and ϕ(σ p ) denotes the equivalent (plastic) stress
that is compared to the yield limit σ y , a material property. Then the evolution law for
the plastic strain (i.e. the associated flow rule) follows alternatively to Eq. 5.94a from
the postulate of maximum dissipation (due to plasticity) with a Lagrange functional
incorporating the admissibility constraint φ ≤ 0 by the Lagrange multiplier λ ≥ 0
(σ p , λ; ˙
p ) := −d(σ p ; ˙
p ) + λ φ(σ p ).
(5.96)
Consequently, the stationarity condition of this constrained optimization problem
reads
˙
p = λ ∂ σ p φ,
(5.97)
subject to the optimality (complementary) conditions in Karush–Kuhn–Tucker form
λ ≥ 0, φ ≤ 0, λ φ = 0.
(5.98)
It shall be noted that collectively Eqs. 5.95, 5.97 and 5.98 are entirely equivalent
statements to Eqs. 5.94a and 5.94b.
As a further interesting aspect the dissipation d = σ p ˙
p shall next be examined
more closely. From Eqs. 5.93a and 5.93b the dissipation d is alternatively expressed
in terms of the dissipation potential π and the dual dissipation potential π
∗ as
d = π(˙ p ) + π
∗
(σ p ) ≥ 0.
(5.99)
However, based on the above introduction of the yield condition φ ≤ 0 the dual
dissipation potential is identified as the indicator function I A of the admissible
domain A